Nonlinear Degenerate Matter-type H-semi-variational Inequalities Of Homogeneous

The field of Inequality problems has seen a considerable development in Mathematics, Mechanics and Engineering Science in a remarkably short time. This is mainly due to the fact that new, very efficient mathematical tools used in the area of Inequality problems, or, more generally, the field of Nonsmooth Mechanics, proved beneficial to the promotion of scientific thought and methodology. In the area of Inequality Problems we can distinguish two main directions: that of Variational Inequalities, which already has a research "life" of about 45 years and is mainly connected with convex energy functions, and that of Hemivariational Inequalities which is more "young". The idea of hemivariational inequalities was born only 25 years ago and is connected with nonconvex energy function.Because of the nonmonotone character of the multi-valued boundary conditions many problems in Mechanics and Engineering Science can't be solved by a convex analysis approach. We are lead to a mathematical model invoving the Clarke subdifferential of a locally Lipschitz functional. Such formulation is called a hemivariational inequality and it allows to deal with many engineering problems involving nonmonotone and multi-valued relations and to give positive answers to unsolved or partially unsolved problems. In many problems of physics one has to solve boundary value problems in periodic media considering equations with highly oscillating coefficients. The main application of homogenization is for the asymptotic analysis of the periodic structures. The most general theory of homogenization is H-convergence which is under the name of G-convergence.In this paper we mainly discuss two kinds of nonlinear parabolic hemivariational inequality respectively. Firstly, we establish existence and uniqueness of solutions to nonlinear single-valued parabolic hemivariational inequality. Using the notions of parabolic G-convergence, we investigate the limit behavior of the sequence of solutions to those hemivariational inequalities. Secondly, we consider the nonlinear multi-valued parabolic hemivariational inequalities. Similarily, we prove an existent and unique theory of solutions to those inequalities and using a general notion of G-convergence we examine the convergence behavior of sequence of solutions to multi-valued parabolic hemivariational inequalities.